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Theorem grpss 14519
Description: Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring  R is a group before we know that it is also a ring. (Theorem rnggrp 15362, on the other hand, requires that we know in advance that  R is a ring.) (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpss.g  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
grpss.r  |-  R  e. 
_V
grpss.s  |-  G  C_  R
grpss.f  |-  Fun  R
Assertion
Ref Expression
grpss  |-  ( G  e.  Grp  <->  R  e.  Grp )

Proof of Theorem grpss
StepHypRef Expression
1 grpss.r . . . 4  |-  R  e. 
_V
2 grpss.f . . . 4  |-  Fun  R
3 grpss.s . . . 4  |-  G  C_  R
4 baseid 13206 . . . 4  |-  Base  = Slot  ( Base `  ndx )
5 opex 4253 . . . . . 6  |-  <. ( Base `  ndx ) ,  B >.  e.  _V
65prid1 3747 . . . . 5  |-  <. ( Base `  ndx ) ,  B >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
7 grpss.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
86, 7eleqtrri 2369 . . . 4  |-  <. ( Base `  ndx ) ,  B >.  e.  G
91, 2, 3, 4, 8strss 13199 . . 3  |-  ( Base `  R )  =  (
Base `  G )
10 plusgid 13259 . . . 4  |-  +g  = Slot  ( +g  `  ndx )
11 opex 4253 . . . . . 6  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  _V
1211prid2 3748 . . . . 5  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
1312, 7eleqtrri 2369 . . . 4  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  G
141, 2, 3, 10, 13strss 13199 . . 3  |-  ( +g  `  R )  =  ( +g  `  G )
159, 14grpprop 14517 . 2  |-  ( R  e.  Grp  <->  G  e.  Grp )
1615bicomi 193 1  |-  ( G  e.  Grp  <->  R  e.  Grp )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {cpr 3654   <.cop 3656   Fun wfun 5265   ` cfv 5271   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505
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